EXERCISES
For more practice, see Extra Practice.
Practice and Problem Solving
A
Practice by Example
Example 1
(page 315)
Example 2
(page 315)
Example 3
(page 316)
Example 4
(pages 316–317)
Divide using long division. Check your answers.
1. (x 2 - 3x - 40) 4 (x + 5)
2. (3x 2 + 7x - 20) 4 (x + 4)
3. (x 3 + 3x2 - x + 2) 4 (x - 1)
4. (2x 3 - 3x 2 - 18x - 8) 4 (x - 4)
5. (9x 3 - 18x2 - x + 2) 4 (3x + 1)
6. (9x 2 - 21x - 20) 4 (x - 1)
7. (x 2 - 7x + 10) 4 (x + 3)
8. (x 3 - 13x - 12) 4 (x - 4)
Determine whether each binomial is a factor of x 3 ± 4 x 2 ± x – 6.
9. x + 1
10. x + 2
11. x + 3
12. x - 3
Divide using synthetic division.
13. (x 3 + 3x 2 - x - 3) 4 (x - 1)
14. (x 3 - 4x 2 + 6x - 4) 4 (x - 2)
15. (x 3 - 7x 2 - 7x + 20) 4 (x + 4)
16. (x 3 - 3x 2 - 5x - 25) 4 (x - 5)
17. (x 3 - 2x 2 - 5x + 6) 4 (x - 1)
18. (-2x 3 + 5x 2 - x + 2) 4 (x + 2)
19. (x 2 + 3) 4 (x - 1)
20. (3x 3 + 17x 2 + 21x - 9) 4 (x + 3)
21. (x 3 + 27) 4 (x + 3)
22. (6x 2 - 8x - 2) 4 (x - 1)
Use synthetic division and the given factor to completely factor each
polynomial function.
23. y = x 3 + 2x 2 - 5x - 6; (x + 1)
24. y = x 3 - 4x 2 - 9x + 36; (x + 3)
25. Geometry Refer to the
diagram. The volume in cubic
inches of the decorative box
can be expressed as the
product of the lengths of its
sides as V(x) = x3 + x2 - 6x.
Write linear expressions with
integer coefficients for the
locker’s length and height.
Example 5
(page 317)
width = x – 2
Use synthetic division and the Remainder Theorem to find P(a).
26. P(x) = x 3 + 4x 2 - 8x - 6; a = -2
27. P(x) = x 3 + 4x 2 + 4x; a = -2
28. P(x) = x 3 - 7x 2 + 15x - 9; a = 3
29. P(x) = x 3 + 7x 2 + 4x; a = -2
31. P(x) = 2x 3 - x 2 + 10x + 5; a = 12
32. P(x) = 2x 3 + 4x 2 - 10x - 9; a = 3 33. P(x) = 2x 4 + 6x 3 + 5x 2 - 45; a = -3
30. P(x) = 6x 3 - x 2 + 4x + 3; a = 3
B
Apply Your Skills
34. Reasoning A polynomial P(x) is divided by a binomial x - a. The remainder
is zero. What conclusion can you draw? Explain.
35. Error Analysis A student represented the product of three linear factors as
x 3 - x 2 - 2x. She used x - 1 as one of the factors. Use division to prove that
the student made an error.
318-320
Chapter 6 Polynomials and Polynomial Functions
36. Open-Ended Write a polynomial division that has a quotient of x + 3 and a
remainder of 2.
Divide.
37. (2x 3 + 9x 2 + 14x + 5) 4 (2x + 1)
38. (x 4 + 3x 2 + x + 4) 4 (x + 3)
39. (x 5 + 1) 4 (x + 1)
40. (x 4 + 4x 3 - x- 4) 4 (x 3 - 1)
41. (3x 4 - 5x 3 + 2x 2 + 3x - 2) 4 (3x - 2)
Determine whether each binomial is a factor of x 3 ± x 2 – 16x – 16.
42. x + 2
43. x - 4
44. x + 1
45. x - 1
46. x - 2
47. x + 4
Use synthetic division to determine whether each binomial is a factor of
3x 3 ± 10x 2 – x – 12.
48. x + 3
49. x - 1
50. x + 2
51. x - 4
Divide using synthetic division.
52. (x 4 - 2x 3 + x 2 + x - 1) 4 (x - 1) 53. (x 4 - 6x 2 - 27) 4 (x + 2)
54. (x 4 - 5x 2 + 4x + 12) 4 (x + 2)
C
Challenge
55. Q x 4 - 92 x 3 + 3x 2 - 12 xR 4 Q x - 12 R
56. Reasoning Divide. Look for patterns in your answers.
a. (x 2 - 1) 4 (x - 1)
b. (x 3 - 1) 4 (x - 1)
d. Using the patterns, factor x 5 - 1.
c. (x 4 - 1) 4 (x - 1)
57. Divide. Look for patterns in your answers.
a. (x 3 + 1) 4 (x + 1)
b. (x 5 + 1) 4 (x + 1)
d. Using the patterns, factor x 9 + 1.
c. (x 7 + 1) 4 (x + 1)
58. Critical Thinking Explain why a polynomial of degree n, divided by a
polynomial of degree 1, yields a quotient of degree n - 1 and a remainder
that is a constant.
59. Use synthetic division to find (x 2 + 4) 4 (x - 2i).
60. Writing Suppose 3, -1, and 4 are zeros of a cubic polynomial function. Sketch
a graph of the function. Could there be more than one graph? Explain.
Lesson 6-3 Dividing Polynomials
318-320
Standardized Test Prep
Multiple Choice
61. What is the remainder when x 2 - 5x + 7 is divided by x + 1?
A. -13
B. -1
C. 1
D. 13
62. Which binomial is NOT a factor of x 3 - x 2 - 17x - 15?
F. x - 5
G. x + 1
H. x + 3
Take It to the NET
Online lesson quiz at
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Short
Response
Web Code: agk-0603
I. x + 5
63. Which of the following, when multiplied by x - 1, results in a cubic
polynomial whose standard form has three terms?
B. x 2 - x
C. x 2 - 1
D. x - 1
A. (x - 1) 2
64. One factor of x 3 - 7x 2 - x + 7 is x - 1. What are all the zeros of the
related polynomial function? Show your work.
Mixed Review
Lesson 6-2
Write a polynomial function in standard form with the given zeros.
65. 3, -5
Lesson 5-6
66. 0, 1, 8
68. 1, multiplicity 4
Simplify each expression.
69. (-4i)(6i)
Lesson 4-6
67. -1, 2, 5
70. (2 + i)(2 - i)
71. (4 - 3i)(5 + i)
Find the inverse of each matrix, if it exists.
21 0 23
72. £ 21 1 20 §
21 0 21
1 2 20
73. £ 0 2 22 §
1 0 22
22 21 20
74. £ 21 21 22 §
23 22 24
Algebra at Work
Quality Control Engineer
Quality control engineers establish procedures for assuring that
products meet minimum standards of quality such as length or
purity. Each product is sampled regularly. Data relating to each
standard are collected and analyzed. All samples must fall within
certain limiting parameters. Quality is further controlled by
requiring that a significant portion of the samples fall within even
stricter limiting parameters.
Take It to the NET For more information about quality
control, go to www.PHSchool.com.
Web Code: agb-2031
318-320
Chapter 6 Polynomials and Polynomial Functions